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A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth
1. | Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense 400, 13566-590 São Carlos - SP, Brazil |
2. | Università degli Studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy |
3. | Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, 22451-900 Gávea - Rio de Janeiro, Brazil |
$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle = \lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n +h_i(x), $ |
$ i = 1, \cdots, n $ |
$ C^{1, 1} $ |
$ \Omega\subset \mathbb{R}^N $ |
$ n\geq 1 $ |
$ \lambda \in \mathbb{R} $ |
$ c_{ij}, \, h_i \in L^\infty(\Omega) $ |
$ c_{ij}\geq 0 $ |
$ M_i $ |
$ 0<\mu_1 I\leq M_i\leq \mu_2 I $ |
$ F_i $ |
References:
[1] |
D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka,
Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268 (2015), 2298-2335.
doi: 10.1016/j.jfa.2015.01.014. |
[2] |
S. N. Armstrong,
Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations, 246 (2009), 2958-2987.
doi: 10.1016/j.jde.2008.10.026. |
[3] |
C. Bandle and W. Reichel, Solutions of quasilinear second-order elliptic boundary value problems via degree theory, in Handbook of Differential Equations (eds. M. Chipot and P. Quittner), Stationary Partial Differential Equations, vol.1. Elsevier, NorthHolland, Amsterdam, (2004), 1–70. Google Scholar |
[4] |
G. Barles, A. Blanc, C. Georgelin and M. Kobylanski,
Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions, Ann. Sc. Norm. Sup. Pisa, 28 (1999), 381-404.
|
[5] |
G. Barles and F. Murat,
Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Rat. Mech. Anal., 133 (1995), 77-101.
doi: 10.1007/BF00375351. |
[6] |
H. Berestycki, L. Nirenberg and S. Varadhan,
The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[7] |
L. Boccardo, F. Murat and J. P. Puel,
Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Sup. Pisa, 11 (1984), 213-235.
|
[8] |
L. Boccardo, F. Murat and J.P. Puel, Existence de solutions faibles des équations elliptiques quasi-lineaires à croissance quadratique, in: Nonlinear P.D.E. and Their Applications (eds. H. Brézis and J.L. Lions), Collège de France Seminar, vol. IV, Research Notes in Mathematics, Pitman, London, 84 (1983), 19–73. |
[9] |
J. Busca and B. Sirakov,
Harnack type estimates for nonlinear elliptic systems and applications, Ann. I. H. Poincaré, 21 (2004), 543-590.
doi: 10.1016/j.anihpc.2003.06.001. |
[10] |
L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995.
doi: 10.1090/coll/043. |
[11] |
L. Caffarelli, M. G. Crandall, M. Kocan and A. Świech,
On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397.
doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. |
[12] |
C. De Coster and L. Jeanjean,
Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient, J. Differential Equations, 262 (2017), 5231-5270.
doi: 10.1016/j.jde.2017.01.022. |
[13] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[14] |
P. Felmer, A. Quaas and B. Sirakov,
Resonance phenomena for second-order stochastic control equations, SIAM Journal on Mathematical Analysis, 42 (2010), 997-1024.
doi: 10.1137/080744268. |
[15] |
V. Ferone and F. Murat,
Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonl. Anal., 42 (2000), 1309-1326.
doi: 10.1016/S0362-546X(99)00165-0. |
[16] |
L. Jeanjean and B. Sirakov,
Existence and multiplicity for elliptic problems with quadratic growth in the gradient, Comm. Part. Diff. Eq., 38 (2013), 244-264.
doi: 10.1080/03605302.2012.738754. |
[17] |
J. L. Kazdan and R. J. Kramer,
Invariant criteria for existence of solutions to second-order quasi-linear elliptic equations, Comm. Pure Appl. Math., 31 (1978), 619-645.
doi: 10.1002/cpa.3160310505. |
[18] |
S. Koike,
Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28.
|
[19] |
S. Koike and A. Świech,
Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan., 61 (2009), 723-755.
doi: 10.2969/jmsj/06130723. |
[20] |
S. Koike and A. Świech,
Local maximum principle for $L^p$-viscosity solutions of fully nonlinear PDEs with unbounded ingredients, Commun. Pure Appl. Anal., 11 (2012), 1897-1910.
doi: 10.3934/cpaa.2012.11.1897. |
[21] |
G. Nornberg and B. Sirakov,
A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient, J. Funct. Anal., 276 (2019), 1806-1852.
doi: 10.1016/j.jfa.2018.06.017. |
[22] |
G. Nornberg,
$C^{1, \alpha}$ regularity for fully nonlinear elliptic equations with superlinear growth in the gradient, J. Math. Pures et Appl., 128 (2019), 297-329.
doi: 10.1016/j.matpur.2019.06.008. |
[23] |
G. Nornberg, Methods of the Regularity Theory in the Study of Partial Differential Equations with Natural Growth in the Gradient, Ph.D. thesis, PUC-Rio, 2018.
doi: 10.17771/PUCRio.acad.36015. |
[24] |
A. Quaas and B. Sirakov,
Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135.
doi: 10.1016/j.aim.2007.12.002. |
[25] |
B. Sirakov,
Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE, International Mathematics Research Notices, 2018 (2018), 7457-7482.
doi: 10.1093/imrn/rnx107. |
[26] |
B. Sirakov,
Solvability of uniformly elliptic fully nonlinear PDE, Archive for Rational Mechanics and Analysis, 195 (2010), 579-607.
doi: 10.1007/s00205-009-0218-9. |
[27] |
B. Sirakov, Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient, preprint, arXiv: 1509.04495. Google Scholar |
[28] |
B. Sirakov, A new method of proving a priori bounds for superlinear elliptic PDE, preprint, arXiv: 1904.03245. Google Scholar |
[29] |
P. Souplet,
A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient, Nonlinear Anal., 121 (2015), 412-423.
doi: 10.1016/j.na.2015.02.005. |
[30] |
N. Winter,
$W^{2, p}$ and $W^{1, p}$ estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.
doi: 10.4171/ZAA/1377. |
show all references
References:
[1] |
D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka,
Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268 (2015), 2298-2335.
doi: 10.1016/j.jfa.2015.01.014. |
[2] |
S. N. Armstrong,
Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations, 246 (2009), 2958-2987.
doi: 10.1016/j.jde.2008.10.026. |
[3] |
C. Bandle and W. Reichel, Solutions of quasilinear second-order elliptic boundary value problems via degree theory, in Handbook of Differential Equations (eds. M. Chipot and P. Quittner), Stationary Partial Differential Equations, vol.1. Elsevier, NorthHolland, Amsterdam, (2004), 1–70. Google Scholar |
[4] |
G. Barles, A. Blanc, C. Georgelin and M. Kobylanski,
Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions, Ann. Sc. Norm. Sup. Pisa, 28 (1999), 381-404.
|
[5] |
G. Barles and F. Murat,
Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Rat. Mech. Anal., 133 (1995), 77-101.
doi: 10.1007/BF00375351. |
[6] |
H. Berestycki, L. Nirenberg and S. Varadhan,
The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[7] |
L. Boccardo, F. Murat and J. P. Puel,
Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Sup. Pisa, 11 (1984), 213-235.
|
[8] |
L. Boccardo, F. Murat and J.P. Puel, Existence de solutions faibles des équations elliptiques quasi-lineaires à croissance quadratique, in: Nonlinear P.D.E. and Their Applications (eds. H. Brézis and J.L. Lions), Collège de France Seminar, vol. IV, Research Notes in Mathematics, Pitman, London, 84 (1983), 19–73. |
[9] |
J. Busca and B. Sirakov,
Harnack type estimates for nonlinear elliptic systems and applications, Ann. I. H. Poincaré, 21 (2004), 543-590.
doi: 10.1016/j.anihpc.2003.06.001. |
[10] |
L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995.
doi: 10.1090/coll/043. |
[11] |
L. Caffarelli, M. G. Crandall, M. Kocan and A. Świech,
On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397.
doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. |
[12] |
C. De Coster and L. Jeanjean,
Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient, J. Differential Equations, 262 (2017), 5231-5270.
doi: 10.1016/j.jde.2017.01.022. |
[13] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[14] |
P. Felmer, A. Quaas and B. Sirakov,
Resonance phenomena for second-order stochastic control equations, SIAM Journal on Mathematical Analysis, 42 (2010), 997-1024.
doi: 10.1137/080744268. |
[15] |
V. Ferone and F. Murat,
Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonl. Anal., 42 (2000), 1309-1326.
doi: 10.1016/S0362-546X(99)00165-0. |
[16] |
L. Jeanjean and B. Sirakov,
Existence and multiplicity for elliptic problems with quadratic growth in the gradient, Comm. Part. Diff. Eq., 38 (2013), 244-264.
doi: 10.1080/03605302.2012.738754. |
[17] |
J. L. Kazdan and R. J. Kramer,
Invariant criteria for existence of solutions to second-order quasi-linear elliptic equations, Comm. Pure Appl. Math., 31 (1978), 619-645.
doi: 10.1002/cpa.3160310505. |
[18] |
S. Koike,
Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28.
|
[19] |
S. Koike and A. Świech,
Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan., 61 (2009), 723-755.
doi: 10.2969/jmsj/06130723. |
[20] |
S. Koike and A. Świech,
Local maximum principle for $L^p$-viscosity solutions of fully nonlinear PDEs with unbounded ingredients, Commun. Pure Appl. Anal., 11 (2012), 1897-1910.
doi: 10.3934/cpaa.2012.11.1897. |
[21] |
G. Nornberg and B. Sirakov,
A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient, J. Funct. Anal., 276 (2019), 1806-1852.
doi: 10.1016/j.jfa.2018.06.017. |
[22] |
G. Nornberg,
$C^{1, \alpha}$ regularity for fully nonlinear elliptic equations with superlinear growth in the gradient, J. Math. Pures et Appl., 128 (2019), 297-329.
doi: 10.1016/j.matpur.2019.06.008. |
[23] |
G. Nornberg, Methods of the Regularity Theory in the Study of Partial Differential Equations with Natural Growth in the Gradient, Ph.D. thesis, PUC-Rio, 2018.
doi: 10.17771/PUCRio.acad.36015. |
[24] |
A. Quaas and B. Sirakov,
Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135.
doi: 10.1016/j.aim.2007.12.002. |
[25] |
B. Sirakov,
Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE, International Mathematics Research Notices, 2018 (2018), 7457-7482.
doi: 10.1093/imrn/rnx107. |
[26] |
B. Sirakov,
Solvability of uniformly elliptic fully nonlinear PDE, Archive for Rational Mechanics and Analysis, 195 (2010), 579-607.
doi: 10.1007/s00205-009-0218-9. |
[27] |
B. Sirakov, Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient, preprint, arXiv: 1509.04495. Google Scholar |
[28] |
B. Sirakov, A new method of proving a priori bounds for superlinear elliptic PDE, preprint, arXiv: 1904.03245. Google Scholar |
[29] |
P. Souplet,
A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient, Nonlinear Anal., 121 (2015), 412-423.
doi: 10.1016/j.na.2015.02.005. |
[30] |
N. Winter,
$W^{2, p}$ and $W^{1, p}$ estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.
doi: 10.4171/ZAA/1377. |
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